Page 281 - 35Linear Algebra

15.1 Review Problems 281

To diagonalize a real symmetric matrix, begin by building an orthogonal
matrix from an orthonormal basis of eigenvectors, as in the example below.

Example 142 The symmetric matrix

2 1
M = ,
1 2

1 1
has eigenvalues 3 and 1 with eigenvectors and respectively. After normal-
1 −1
izing these eigenvectors, we build the orthogonal matrix:
1 1 !
√ √
2 2
P = .
√
√ 1 −1
2 2
T
Notice that P P = I. Then:
3 1 ! 1 1 ! !
√ √ √ √ 3 0
2 2 2 2
MP = = .
√
√
√ 3 −1 √ 1 −1 0 1
2 2 2 2
T
In short, MP = PD, so D = P MP. Then D is the diagonalized form of M
and P the associated change-of-basis matrix from the standard basis to the basis of
eigenvectors.
3 × 3 Example

15.1 Review Problems

Reading Problems 1 , 2 ,
Webwork:
Diagonalizing a symmetric matrix 3, 4

1. (On Reality of Eigenvalues)
√
(a) Suppose z = x + iy where x, y ∈ R, i = −1, and z = x − iy.
Compute zz and zz in terms of x and y. What kind of numbers
are zz and zz? (The complex number z is called the complex
conjugate of z).
(b) Suppose that λ = x + iy is a complex number with x, y ∈ R, and
that λ = λ. Does this determine the value of x or y? What kind
of number must λ be?

281